Finite Group Databasebeta

← #100821

Subgroups — #100821

Order 512.

392 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29113C211yesmaximal, characteristic
2#21907C211yesmaximal, characteristic
3#7708C211yesmaximal, characteristic, complemented
4#7707C211yesmaximal, characteristic, complemented
5#7645C211yesmaximal, characteristic
6#7046C211yesmaximal, characteristic, complemented
7#7046C211yesmaximal, characteristic, complemented
8#2857C2211yescharacteristic
9#2297C2211yescharacteristic
10#2072C411yescharacteristic
11#2071C411yescharacteristic
12#2024C411yescharacteristic
13#2023C411yescharacteristic
14#1936—21no—
15#1935—21no—
16#1924—22no—
17#1924—22no—
18#1825—21no—
19#1817C2211yescharacteristic
20#1817C2211yescharacteristic
21#1798—21no—
22#1788—21no—
23#1780—21no—
24#1749C2211yescharacteristic
25#1749C2211yescharacteristic
26#1749C2211yescharacteristic
27#1272—21no—
28#1272—22no—
29#1271—22no—
30#1270—21no—
31C22×#41D411yescharacteristic
32C2×C4⋊Q8C2×C411yescharacteristic
33C2×C22⋊Q8—21no—
34C2×#117D411yescharacteristic
35C22×C42C2311yescharacteristic
36C2×C4⋊C8—22no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×#97—21no—
49C2×#97—21no—
50C2×#97—21no—
51#391—21no—
52#391C2×C411yescharacteristic
53#391D411yescharacteristic
54#390C2×C411yescharacteristic
55#390D411yescharacteristic
56#389—21no—
57#389—21no—
58#386—21no—
59#386—21no—
60#384C2×C411yescharacteristic
61#380—21no—
62#380—21no—
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65#340—21no—
66#340—21no—
67#340—21no—
68#340—21no—
69#336—22no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336C2×C411yescharacteristic
75#336C2×C411yescharacteristic
76#336D412yes—
77#336D412yes—
78C2×#41—41no—
79C2×#41—41no—
80C2×#41—42no—
81C2×#41—81no—
82C22×Q8—21no—
83C22×Q8C22⋊C411yescharacteristic
84C22×D4—21no—
85C22×D4C22⋊C411yescharacteristic
86C23×C4—21no—
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—41no—
95C22×C8—41no—
96C4⋊Q8—81no—
97C22⋊Q8—82no—
98#117—42no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C22⋊C411yescharacteristic
106C2×C4⋊C4C22⋊C411yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—41no—
113C2×C42—21no—
114C2×C42—81no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C4⋊C8—82no—
120#103—42no—
121#103—81no—
122#102—42no—
123#102—81no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—42no—
129C22⋊C8—82no—
130C22⋊C8—82no—
131C22⋊C8—81no—
132C22⋊C8—81no—
133#97—81no—
134#97—81no—
135#97—81no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—41no—
141#95—41no—
142C24—21no—
143C24#12311yescharacteristic
144#41—84no—
145#41—84no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—42no—
158C2×D4—81no—
159C2×D4—81no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4#12011yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—82no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C22⋊C4—82no—
218C42—41no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—42no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23#38611yescharacteristic
238Q8—42no—
239Q8—81no—
240Q8—162no—
241D4—42no—
242D4—81no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4#42111yescharacteristic
281C2×C4#42111yescharacteristic
282C2×C4#45512yes—
283C8—82no—
284C8—82no—
285C8—81no—
286C8—81no—
287C8—161no—
288C8—161no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#188111yescharacteristic
304C22#188111yescharacteristic
305C22#188511yescharacteristic
306C22#189711yescharacteristic
307C22#190211yescharacteristic
308C22#190311yescharacteristic
309C22#191311yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—161no—
321C4—161no—
322C2—81no—
323C2—161no—
324C2#900511yescharacteristic
325C2#931311yescharacteristic
326C2#931411yescharacteristic
327C2#933111yescharacteristic
328C2#933211yescharacteristic
329C2#941111yescharacteristic
330C2#941911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #7708C2semidirect
4. #7707C2semidirect
6. #7046C2semidirect
7. #7046C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#291132
#219072
#77082
#77072
#76452
#70462